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• W3127030265 abstract "The main theorem of this PhD thesis states the following: the genus 0 coefficients of the quantum Witten- Kontsevich series defined by Buryak, Dubrovin, Guere, and Rossi are equal to the coefficients of the polynomials defined by Goulden, Jackson, and Vakil in their study of double Hurwitz numbers. We also prove several other results on the quantum Witten-Kontsevich series. The classical Witten-Kontsevich series is a generating series of intersection numbers on the moduli spaces of stable curves. The Witten conjecture, proved by Kontsevich, asserts that this series is the logarithm of a tau function of the KdV hierarchy. In 2016, Buryak and Rossi introduced a new way to construct quantum integrable hierarchies, including a quantum KdV hierarchy. Buryak, Dubrovin, Guere and Rossi then defined quantum tau functions, one of which is the quantum Witten-Kontsevich series. This series depends on two parameters: the genus parameter _ and the quantization parameter ~. It reduces to the Witten-Kontsevich series when we plug ~ = 0. One-part double Hurwitz numbers count non-equivalent holomorphic maps from a Riemann surface of genus g to P1 with a prescribed ramification profile over 0, a complete ramification over 1, and a given number of simple ramifications elsewhere. Goulden, Jackson and Vakil proved that these numbers are polynomial in the orders of ramification over 0. We show that the coefficients of these polynomials are equal to the coefficients of the quantum Witten-Kontsevich series with _ = 0. In Chapter 1, we present the setting of the classical and quantum integrable hierarchies that we will use. We also present the construction of classical and quantum tau functions. In Chapter 2, we present the moduli spaces of curves Mg;n and their tautological rings. We briefly review the Witten conjecture. Then we introduce the double ramification cycle and discuss various methods for computing it. This cycle is needed to define the Hamiltonians of quantum integrable hierarchies. In Chapter 3, we present the quantum KdV hierarchy and some of its properties. We then define the quantum Witten-Kontsevich series as a particular quantum tau function of this hierarchy. In Chapter 4, we introduce Hurwitz numbers. We first present a remarkable link between the quantum KdV hierarchy and the cut-and-join equation. Then we introduce the so-called one-part double Hurwitz numbers. Their relation with the quantum Witten-Kontsevich series is the main result ot this thesis. In Chapter 5, we present Eulerian numbers. These numbers appear in the computations of the coefficients of the quantum Witten-Kontsevich series. Their properties are crucial for our proofs. In Chapter 6, we formulate and prove our main theorem and other results on the quantum Witten- Kontsevich series." @default.
• W3127030265 created "2021-02-15" @default.
• W3127030265 creator A5069419574 @default.
• W3127030265 date "2020-07-03" @default.
• W3127030265 modified "2023-09-23" @default.
• W3127030265 title "The quantum Witten-Kontsevich series." @default.
• W3127030265 hasPublicationYear "2020" @default.
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